On the Min4PC Matrix of a Tree

Abstract

The Four point condition (abbreviated as 4PC) is a condition used to test if a given distance matrix arises from shortest path distances on trees. From a tree \(T\), Bapat and Sivasubramanian defined a matrix \(\operatorname{Min4PC}_T\) based on this condition. They also gave a basis \(B\) for the row space of \(\operatorname{Min4PC}_T\) and determined its Smith Normal Form. In this paper, we consider the matrix \(\operatorname{Min4PC}_T[B,B]\) restricted to a basis \(B\) and give an explicit inverse for it. It is known that the distance matrix \(D_T\) of a tree \(T\), is invertible and that its inverse is a rank-one update of its scaled Laplacian. Our inverse has a similar form and is a rank-one update of a Laplacian like matrix.